Cambridge Additional Mathematics

Functions (Chapter 2) 51

The domain of the composite of two functions depends on the domain of the original functions.

For example, consider f(x)=x^2 with domain x 2 R and g(x)=

p

x with domain x> 0.

(f±g)(x)=f(g(x))

=(

p

x)^2

=x

The domain of (f±g)(x) is x> 0 , notR, since (f±g)(x)

is defined using function g(x).

EXERCISE 2E

1 Given f:x 7! 2 x+3 and g:x 7! 1 ¡x, find in simplest form:

a (f±g)(x) b (g±f)(x) c (f±g)(¡3)

2 Given f(x)=2+x and g(x)=3¡x, find:

a fg(x) b gf(x) c f^2 (x)

3 Given f(x)=

p

6 ¡x and g(x)=5x¡ 7 , find:

a (g±g)(x) b (f±g)(1) c (g±f)(6)

4 Given f:x 7 !x^2 and g:x 7! 2 ¡x, find (f±g)(x) and (g±f)(x).

Find also the domain and range of f±g and g±f.

5 Suppose f(x)=3x+5 and g(x)=2x¡ 3.

a Find (f±g)(x). b Solve (f±g)(x)=g(x¡2).

6 Suppose f:x 7 !x^2 +1 and g:x 7! 3 ¡x.

a Find in simplest form: i fg(x) ii gf(x)

b Find the value(s) ofxsuch that gf(x)=f(x).

7aIf ax+b=cx+d for all values ofx, show that a=c and b=d.

Hint: If it is true for allx, it is true for x=0and x=1.

b Given f(x)=2x+3 and g(x)=ax+b and that (f±g)(x)=x for all values ofx,

deduce that a=^12 and b=¡^32.

c Is the result inbtrue if (g±f)(x)=x for allx?

8 Given f(x)=

p

1 ¡x and g(x)=x^2 , find:

a (f±g)(x) b the domain and range of (f±g)(x).

Sometimes we do not wish to draw a time-consuming graph of a function but wish to know when the

function is positive, negative, zero, or undefined. Asign diagramenables us to do this and is relatively

easy to construct.

For the function f(x), the sign diagram consists of:

² ahorizontal linewhich is really thex-axis

² positive(+) andnegative(¡) signs indicating that the graph isaboveandbelowthex-axis respectively

² thezerosof the function, which are thex-intercepts of the graph of y=f(x), and therootsof the

equation f(x)=0

² values ofxwhere the graph is undefined.

F Sign diagrams

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_02\051CamAdd_02.cdr Thursday, 3 April 2014 4:08:58 PM BRIAN